Well actually most DSLRs have more than one filter. Yes there is an IR Cut filter that Aidan refers to below, but there is usually a blue cast filter (sometimes referred to as LP2) that skews the camera towards producing good daytime colour. The Bayer Matrix does stop alot of the red recognition, due to the preference in the matrix design given to blue and green pixel filter layers. If you look at the illustration here:
https://en.wikipedia.org/wiki/Bayer_filter
it shows you what I mean.
Doing an Astro filter mod on a stock DSLR will open it up to more Ha light but your still relying on a small number of Ha capable pixels to capture it.
A full mono conversion of a DSLR (a De-bayering of the sensor) will open it up to narrowband imaging, where every pixel can capture Ha with the appropriate filter on the front. However, it will be useless as a standard camera.